2021 AMC 12B Spring Problem 5

Below is the professionally curated solution for Problem 5 of the 2021 AMC 12B Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 12B Spring solutions, or check the answer key.

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Concepts:transformation work backwards

Difficulty rating: 1330

The point $P(a,b)$ in the $xy$ -plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y=-x.$ The image of $P$ after these two transformations is at $(-6,3).$ What is $b-a?$

$1$

$3$

$5$

$7$

$9$

Solution:

A $90^\circ$ counterclockwise rotation about $(1,5)$ sends $(a,b)$ to $(1-(b-5),\,5+(a-1))=(6-b,\,4+a).$

Reflecting that about $y=-x$ (which maps $(x,y)$ to $(-y,-x)$ ) gives $(-(4+a),\,-(6-b))=(-4-a,\,b-6).$

Setting this equal to $(-6,3)$ gives $-4-a=-6$ and $b-6=3,$ so $a=2$ and $b=9.$

Therefore $b-a=9-2=7.$

Thus, the correct answer is D.

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